What are the precise definitions of "Euclidean space" and "Space"? What is the precise definitions of "Euclidean space" and "Space"?
This question was asked in my viva-exam i'm not sure about my answer-I answered like-"Euclidean space is a space where the notion of Euclidean distance is present",then the examiner counter questioned me "What is a space?" to this i've no answer.
I need to know how should i answered these questions.I've also gone through wikipedia for "Euclidean space",but i did'nt get any precise answer.
Any suggestions are heartly welcome!!
 A: The answer really depends on the context. If the question was in the context of a Euclidean geometry class and was about the Euclidean plane, one could give Hilbert's list of axioms, including his primitive notions. (I am actually not even sure about a reference for a Hilbert-type list of axioms for the Euclidean 3-space.) If it was in the context of a linear algebra class, the right answer would be to give axioms of an $n$-dimensional real vector space equipped with an inner product, but regarded as an affine space rather than a vector space (i.e. undergone the "affinization"). If this were in the context of a metric geometry class, one would give yet another definition and state that the $n$-dimensional Euclidean space $E^n$ is a metric space  satisfying a certain list of properties which I do not want to give here. Lastly, if this were in the context of a differential geometry class, the answer would be that $E^n$ is the (isometry class of) the $n$-dimensional complete simply connected Riemannian manifold of zero curvature. 
